Finite Group Databasebeta

← #10690410

Subgroups — #10690410

Order 768.

538 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3169C211yesmaximal, characteristic
2#83277C211yesmaximal, characteristic, complemented
3#71818C211yesmaximal, characteristic, complemented
4#71811C211yesmaximal, characteristic, complemented
5#71800C211yesmaximal, characteristic, complemented
6#67656C211yesmaximal, characteristic, complemented
7#67655C211yesmaximal, characteristic, complemented
8#9053—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C3×#532C2211yescharacteristic
11C2×#948—21no—
12C3×#500C411yescharacteristic, complemented
13C3×#493C411yescharacteristic, complemented
14C3×#486C411yescharacteristic, complemented
15C3×#486C411yescharacteristic, complemented
16C3×#466—22no—
17C3×C4×D8—22no—
18C2×C3×C4×C8C2211yescharacteristic
19C2×#837—21no—
20C2×#837—21no—
21#5234—21no—
22#5012—21no—
23#5011—22no—
24D4×C3⋊C8—21no—
25C4×C4⋊Dic3C2211yescharacteristic
26#4925C2211yescharacteristic
27C2×C4⋊(C3⋊C8)C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4537—22no—
30D4⋊(C3⋊C8)—21no—
31D4⋊(C3⋊C8)—21no—
32#3169D311yescharacteristic
33#2935—31no—
34#1980—31no—
35#1502—31no—
36#1480—31no—
37#1279—31no—
38#1278—31no—
39C2×C3×#41—21no—
40C2×C3×#41—21no—
41C2×C3×D8—22no—
42C22×C3×C8—21no—
43C3×C4⋊Q8C2×C411yescharacteristic
44C3×C4⋊Q8C2×C411yescharacteristic
45C3×#127C2×C411yescharacteristic
46C3×#127C2×C411yescharacteristic
47C3×#124—21no—
48C3×#124—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×#121D412yes—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×C4×D4D412yes—
57C2×C3×C42C2311yescharacteristic
58#948—42no—
59#948—42no—
60C2×C4⋊Dic3—21no—
61C2×C4×Dic3—21no—
62C2×#208—22no—
63C2×#208D411yescharacteristic
64C2×#208D411yescharacteristic
65C22×C3⋊C8—22no—
66C22×C3⋊C8—21no—
67C3×#107—22no—
68C3×#102—22no—
69C3×#102—22no—
70C3×C4×C8—21no—
71C3×C4×C8C2×C411yescharacteristic
72C3×C4×C8C2×C411yescharacteristic
73#837—42no—
74#837—42no—
75C3⋊#95—21no—
76#830—22no—
77#830—22no—
78#830—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)D411yescharacteristic
82C4⋊(C3⋊C8)D411yescharacteristic
83#803—21no—
84C4×C3⋊C8—21no—
85C4×C3⋊C8—21no—
86C2×#131—61no—
87#532C2×D311yescharacteristic
88#532C2×D311yescharacteristic
89#500Dic311yescharacteristic, complemented
90#493Dic311yescharacteristic, complemented
91#486Dic311yescharacteristic, complemented
92#486Dic311yescharacteristic, complemented
93#466—22no—
94C4×D8—22no—
95#436—61no—
96#435—62no—
97C8×D4—61no—
98#431—31no—
99C2×C4⋊C8—31no—
100C2×#104—61no—
101C2×#104—61no—
102#408—61no—
103C4×M16—31no—
104C2×C4×C8C2×D311yescharacteristic
105C4×C4⋊C4—31no—
106#327—62no—
107D4⋊C8—61no—
108D4⋊C8—61no—
109C3×#41—41no—
110C3×#41—41no—
111C3×#41—41no—
112C3×#41—41no—
113C3×#41—41no—
114C3×#41—41no—
115C3×#41—41no—
116C3×#41—41no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×D4—22no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—41no—
126C2×C3×D4C22⋊C412yes—
127C22×C3×C4—22no—
128C22×C3×C4—41no—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C22×Dic3—41no—
133C3×D8—44no—
134C2×C3×C8—21no—
135C2×C3×C8—21no—
136C2×C3×C8—42no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4C22⋊C412yes—
141C3×C22⋊C4—22no—
142C3×C22⋊C4—22no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C42C22×C411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42#4111yescharacteristic
149C4⋊Dic3—22no—
150C4⋊Dic3—41no—
151C4×Dic3—41no—
152C4×Dic3—41no—
153#208—21no—
154#208—21no—
155#208—21no—
156#208—44no—
157#208D812yescomplemented
158C2×C3⋊C8—21no—
159C2×C3⋊C8—21no—
160C2×C3⋊C8—21no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—44no—
163C2×C3⋊C8—41no—
164C2×C3⋊C8—41no—
165C2×C3⋊C8—41no—
166C2×#41—21no—
167C2×#41—21no—
168C2×D8—22no—
169#131—122no—
170#131—122no—
171C2×M16—31no—
172C2×M16—31no—
173C2×M16—62no—
174C22×C8—21no—
175C22×C8—62no—
176C22×C8—61no—
177C4⋊Q8C2×Dic311yescharacteristic
178C4⋊Q8C2×Dic311yescharacteristic
179#127C2×Dic311yescharacteristic
180#127C2×Dic311yescharacteristic
181#124—21no—
182#124—21no—
183#121—21no—
184#121—21no—
185#121—21no—
186#121#6712yes—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4#6712yes—
191C2×C4⋊C4—61no—
192C2×C42—61no—
193C2×C42C22×D311yescharacteristic
194#107—22no—
195C4⋊C8—31no—
196C4⋊C8—31no—
197C4⋊C8—61no—
198C4⋊C8—61no—
199#104—122no—
200#104—122no—
201#102—22no—
202#102—22no—
203C22⋊C8—62no—
204C22⋊C8—62no—
205C22⋊C8—61no—
206#97—61no—
207C4×C8—21no—
208C4×C8—61no—
209C4×C8—61no—
210C4×C8C2×Dic311yescharacteristic
211C4×C8C2×Dic311yescharacteristic
212#95—61no—
213C23×C3—22no—
214C23×C3—41no—
215C23×C3#12111yescharacteristic
216C3×Q8—41no—
217C3×Q8—41no—
218C3×Q8—41no—
219C3×Q8—41no—
220C3×D4—42no—
221C3×D4—42no—
222C3×D4—42no—
223C3×D4—42no—
224C3×D4—41no—
225C3×D4—41no—
226C3×D4—41no—
227C3×D4—41no—
228C3×D4—41no—
229C3×D4—41no—
230C3×D4—41no—
231C3×D4—41no—
232C3×D4—81no—
233C3×D4—81no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—22no—
237C2×C3×C4—41no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—41no—
241C2×C3×C4—81no—
242C2×C3×C4—81no—
243C2×C3×C4C2×C22⋊C411yescharacteristic
244C2×C3×C4C4×D411yescharacteristic
245C2×C3×C4C4×D411yescharacteristic
246C2×C3×C4#12011yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#12311yescharacteristic
249C2×C3×C4C2×D811yescharacteristic
250C2×C3×C4#13511yescharacteristic
251C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
252C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
253C2×Dic3—42no—
254C2×Dic3—81no—
255C2×Dic3—81no—
256C3×C8—22no—
257C3×C8—41no—
258C3⋊C8—22no—
259C3⋊C8—41no—
260C3⋊C8—41no—
261C3⋊C8—41no—
262C3⋊C8—81no—
263#41—41no—
264#41—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268#41—41no—
269#41—41no—
270#41—41no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×D4—22no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—41no—
280C2×D4#21712yes—
281C22×C4—22no—
282C22×C4—41no—
283C22×C4—121no—
284C22×C4D3×D411yescharacteristic
285C22×C4#23711yescharacteristic
286C22×C4C2×#6711yescharacteristic
287D8—44no—
288M16—32no—
289M16—61no—
290M16—61no—
291M16—61no—
292M16—124no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—42no—
296C2×C8—61no—
297C2×C8—61no—
298C2×C8—61no—
299C2×C8—61no—
300C2×C8—124no—
301C2×C8—121no—
302C2×C8—121no—
303C2×C8—121no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—62no—
308C4⋊C4—121no—
309C4⋊C4#21712yes—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C42—121no—
315C42—121no—
316C42D3×D411yescharacteristic
317C42#23711yescharacteristic
318C42C22×Dic311yescharacteristic
319C42C2×#6711yescharacteristic
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—42no—
323C22×C3—42no—
324C22×C3—42no—
325C22×C3—81no—
326C22×C3—81no—
327C22×C3—81no—
328C22×C3#38611yescharacteristic
329C22×C3#45911yescharacteristic
330C22×C3#46311yescharacteristic
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—42no—
335C3×C4—81no—
336C3×C4C4×D811yescharacteristic
337C3×C4#44211yescharacteristic
338C3×C4#44311yescharacteristic
339C3×C4#44411yescharacteristic
340C3×C4#44711yescharacteristic
341C3×C4#44911yescharacteristic
342Dic3—82no—
343C23—22no—
344C23—41no—
345C23#93911yescharacteristic
346Q8—41no—
347Q8—41no—
348Q8—41no—
349Q8—41no—
350D4—42no—
351D4—42no—
352D4—42no—
353D4—42no—
354D4—41no—
355D4—41no—
356D4—41no—
357D4—41no—
358D4—41no—
359D4—41no—
360D4—41no—
361D4—41no—
362D4—81no—
363D4—81no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—122no—
374C2×C4—241no—
375C2×C4—241no—
376C2×C4D3×D811yescharacteristic
377C2×C4#91111yescharacteristic
378C2×C4#91111yescharacteristic
379C2×C4#91211yescharacteristic
380C2×C4D4×Dic311yescharacteristic
381C2×C4D4×Dic311yescharacteristic
382C2×C4#93511yescharacteristic
383C2×C4#93711yescharacteristic
384C2×C4#93911yescharacteristic
385C2×C4C2×#21711yescharacteristic
386C8—22no—
387C8—41no—
388C8—62no—
389C8—121no—
390C8—121no—
391C8—121no—
392C8—241no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—82no—
396C2×C3—161no—
397C2×C3#187611yescharacteristic
398C2×C3#187911yescharacteristic
399C2×C3#188411yescharacteristic
400C22—21no—
401C22—21no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22#515011yescharacteristic
409C22#515311yescharacteristic
410C22#522011yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—42no—
415C4—81no—
416C4—242no—
417C4Dic3×D811yescharacteristic
418C4#515211yescharacteristic
419C4#515611yescharacteristic
420C4#515811yescharacteristic
421C4#519611yescharacteristic
422C4#519711yescharacteristic
423C3#905311yescharacteristic, complemented, Sylow
424C2—21no—
425C2—21no—
426C2—82no—
427C2—161no—
428C2#7181711yescharacteristic
429C2#7203711yescharacteristic
430C2#7208711yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3169C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3169D3
hypocenterC3#9053
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3169

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #83277C2semidirect
3. #71818C2semidirect
4. #71811C2semidirect
5. #71800C2semidirect
6. #67656C2semidirect
7. #67655C2semidirect
12. C3×#500C4semidirect
13. C3×#493C4semidirect
14. C3×#486C4semidirect
15. C3×#486C4semidirect
89. #500Dic3semidirect
90. #493Dic3semidirect
91. #486Dic3semidirect
92. #486Dic3semidirect
157. #208D8semidirect
423. C3#9053semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#31692
#832772
#718182
#718112
#718002
#676562
#676552
#90533